Orthogonal Polynomials for Area-Type Measures and Image Recovery

E. B. Saff, H. Stahl, N. Stylianopoulos, V. Totik
2015 SIAM Journal on Mathematical Analysis  
Let G be a finite union of disjoint and bounded Jordan domains in the complex plane, let K be a compact subset of G and consider the set G ⋆ obtained from G by removing K; i.e., G ⋆ := G \ K. We refer to G as an archipelago and G ⋆ as an archipelago with lakes. Denote by {pn(G, z)} ∞ n=0 and {pn(G ⋆ , z)} ∞ n=0 , the sequences of the Bergman polynomials associated with G and G ⋆ , respectively; that is, the orthonormal polynomials with respect to the area measure on G and G ⋆ . The purpose of
more » ... . The purpose of the paper is to show that pn(G, z) and pn(G ⋆ , z) have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for G ⋆ are determined by the boundary of G. As a consequence we can analyze certain asymptotic properties of pn(G ⋆ , z) by using the corresponding results for pn (G, z), which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.
doi:10.1137/14096205x fatcat:g67vl7bbhvfqfdn62vdeh6drte